MCQ
$\text{ABC}$ is a right angled triangle, right angled at B such that $BC = 6\ cm$ and $AB = 8\ cm$. A circle with centre $O$ is inscribed in $\triangle ABC$. The radius of the circle is:
- A$1\ cm$
- ✓$2\ cm$
- C$3\ cm$
- D$4\ cm$
✓
Answer
Correct option: B.
$2\ cm$
In a right $\triangle\text{ABC},$ $\angle\text{B}=90^{\circ}$
$BC = 6\ cm$, $AB = 8\ cm$
$AC^2=AB+BC^2 \text { (Pythagoras Theorem) }$
$BC = 6\ cm$, $AB = 8\ cm$
$AC^2=AB+BC^2 \text { (Pythagoras Theorem) }$
$=(8)^2+(6)^2=64+36=100=(10)^2$
$AC=10 cm$
An incircle is drawn with centre $0$ which touches the sides of the triangle $A B C$ at $P, Q$ and $\text{R O P, O Q}$ and $O R$ are radii and $\text{A B, B C}$ an $C A$ are the tangents to the circle.
$O P \perp A B, O Q \perp B C$ and $O R \perp C A$
$\text{OPBQ}$ is a square.
Let $r$ be the radius of the incircle.
$P B=B Q=r$
$A R=A P=8-r,$
$C Q=C R=6-r$
$A C=A R+C R$
$\Rightarrow 10=8-r+6-r$
$\Rightarrow 10=14-2 r$
$\Rightarrow 2 r=14-10=4$
$\Rightarrow r=2$
Radius of the incircle $=2 \ cm$.
$AC=10 cm$
An incircle is drawn with centre $0$ which touches the sides of the triangle $A B C$ at $P, Q$ and $\text{R O P, O Q}$ and $O R$ are radii and $\text{A B, B C}$ an $C A$ are the tangents to the circle.
$O P \perp A B, O Q \perp B C$ and $O R \perp C A$
$\text{OPBQ}$ is a square.
Let $r$ be the radius of the incircle.
$P B=B Q=r$
$A R=A P=8-r,$
$C Q=C R=6-r$
$A C=A R+C R$
$\Rightarrow 10=8-r+6-r$
$\Rightarrow 10=14-2 r$
$\Rightarrow 2 r=14-10=4$
$\Rightarrow r=2$
Radius of the incircle $=2 \ cm$.
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